Course details
Selected parts from mathematics II.
XPC-VPM FEKT XPC-VPM Acad. year 2025/2026 Summer semester 5 credits
The aim of this course is to introduce the basics of calculation of improper multiple integral and basics of solving of linear differential equations using delta function and weighted function.
In the field of improper multiple integral, main attention is paid to calculations of improper multiple integrals on unbounded regions and from unbounded functions.
In the field of linear differential equations, the following topics are covered: Eliminative solution method, method of eigenvalues and eigenvectors, method of variation of constants, method of undetermined coefficients, stability of solutions.
Guarantor
Course coordinator
Language of instruction
Completion
Time span
- 26 hrs lectures
- 26 hrs exercises
Department
Learning objectives
The aim of this course is to introduce the basics of improper multiple integrals, systems of differential equations including of investigations of a stability of solutions of differential equations and applications of selected functions with solving of dynamical systems.
Students completing this course should be able to:
- calculate improper multiple integral on unbounded regions and from unbounded functions.
- apply a weighted function and a delta function to solving of linear differential equations.
- select an optimal solution method for given differential equation.
- investigate a stability of solutions of systems of differential equations.
Prerequisite knowledge and skills
The student should be able to apply the basic knowledge of analytic geometry and mathamatical analysis on the secondary school level: to explain the notions of general, parametric equations of lines and surfaces and elementary functions.
From the BMA1 and BMA2 courses, the basic knowledge of differential and integral calculus and solution methods of linear differential equations with constant coefficients is demanded. Especially, the student should be able to calculate derivative (including partial derivatives) and integral of elementary functions.
Syllabus of lectures
- Impulse functions, solving differential equations using a weight function
- Systems of differential equations, Elimination method
- Constants variation method , Method of eigenvalue numbers and eigenvalue vectors
- Method of undetermined coefficients
- Differential transform method (DTM)
- DTM for systems of differential equations, delayed systems
- Difference equations, rules for differences, summation
- Solving linear homogeneous and non-homogeneous difference equations
- Gamma function, solving specific nonlinear difference equations
- Solving systems of difference equations
- Fractional calculus, Mittag-Leffler functions
- Solving fractional differential equations in the sense of Caputo a Riemann-Liouville derivative
- Solving fractional systems of differential equations, impulse characterizations
Syllabus of numerical exercises
- Impulse functions, solving differential equations using a weight function
- Systems of differential equations, Elimination method
- Constants variation method , Method of eigenvalue numbers and eigenvalue vectors
- Method of undetermined coefficients
- Differential transform method (DTM)
- DTM for systems of differential equations, delayed systems
- Difference equations, rules for differences, summation
- Solving linear homogeneous and non-homogeneous difference equations
- Gamma function, solving specific nonlinear difference equations
- Solving systems of difference equations
- Fractional calculus, Mittag-Leffler functions
- Solving fractional differential equations in the sense of Caputo a Riemann-Liouville derivative
- Solving fractional systems of differential equations, impulse characterizations
Progress assessment
The student's work during the semestr (written tests and homework) is assessed by maximum 30 points.
Written examination is evaluated by maximum 70 points. It consist of seven tasks (one from improper multiple integral (10 points), three from application of a weighted function and a delta function (3 X 10 points) and three from analytical solution method of differential equations (3 x 10 points)).
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.
Course inclusion in study plans
- Programme BIT, 2nd year of study, Elective
- Programme BIT (in English), 2nd year of study, Elective